advanced_notions:poisson_bracket

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advanced_notions:poisson_bracket [2018/04/08 13:38] jakobadmin [Concrete] |
advanced_notions:poisson_bracket [2018/12/18 14:00] (current) jakobadmin |
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<tabbox Concrete> | <tabbox Concrete> | ||

- | that | + | |

For two sets of canonical coordinates | For two sets of canonical coordinates | ||

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This is one way to make the difference between quantum and classical mechanics explicit: | This is one way to make the difference between quantum and classical mechanics explicit: | ||

- | $$ \text{Commutator}\quad [\hat{f},\hat{g}] \quad\longleftrightarrow\quad\text{Poisson bracket}\quad \{f,g\}$$ | + | $$ \text{Commutator}\quad [\hat{f},\hat{g}] \quad\longleftrightarrow\quad\text{Poisson bracket}\quad \{f,g\}$$ |

---- | ---- | ||

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\end{align} | \end{align} | ||

- | which is extremely similar to the [[equations:canonical_commutation_relations|canonical commutation relations]] in quantum mechanics: | + | which is extremely similar to the [[formulas:canonical_commutation_relations|canonical commutation relations]] in quantum mechanics: |

\begin{align} | \begin{align} | ||

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\end{align} | \end{align} | ||

- | In words both sets of equations state that that $p_i$ generates infinitesimal translations of $q_i$. | + | In words both sets of equations state that $p_i$ generates infinitesimal translations of $q_i$. |

---- | ---- | ||

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---- | ---- | ||

- | Any system in [[theories:classical_mechanics|Classical Mechanics]] can be thought of rigorously as a [[basic_tools:phase_space|phase space]] which is more precisely formalized as a [[advanced_tools:symplectic_structure|symplectic]] manifold $(X,ω)$, or even more precisely a Poisson Manifold. In words, this means that the algebra of all functions on our phase space $X$, is canonically equipped with a Lie bracket: the Poisson bracket. Formulated differently, dynamics in mechanics are modeled on the cotangent bundle $T^∗M$ which has a canonical symplectic structure. | + | Any system in [[theories:classical_mechanics:newtonian|Classical Mechanics]] can be thought of rigorously as a [[basic_tools:phase_space|phase space]] which is more precisely formalized as a [[advanced_tools:symplectic_structure|symplectic]] manifold $(X,ω)$, or even more precisely a Poisson Manifold. In words, this means that the algebra of all functions on our phase space $X$, is canonically equipped with a Lie bracket: the Poisson bracket. Formulated differently, dynamics in mechanics are modeled on the cotangent bundle $T^∗M$ which has a canonical symplectic structure. |

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<tabbox Why is it interesting?> | <tabbox Why is it interesting?> | ||

- | Poisson brackets are necessary to describe the time evolution of observables in the [[frameworks:hamiltonian_formalism|Hamiltonian formulation]] of [[theories:classical_mechanics|classical mechanics]]. Formulated differently, the Poisson bracket controls the dynamics in classical mechanics. | + | Poisson brackets are necessary to describe the time evolution of observables in the [[formalisms:hamiltonian_formalism|Hamiltonian formulation]] of [[theories:classical_mechanics:newtonian|classical mechanics]]. Formulated differently, the Poisson bracket controls the dynamics in classical mechanics. |

+ | Poisson brackets play more or less the same role in [[theories:classical_mechanics:newtonian|classical mechanics]] that [[formulas:canonical_commutation_relations|commutators]] do in [[theories:quantum_mechanics:canonical|quantum mechanics]]. | ||

- | Poisson brackets play more or less the same role in [[theories:classical_mechanics|classical mechanics]] that [[equations:canonical_commutation_relations|commutators]] do in [[theories:quantum_mechanics|quantum mechanics]]. | + | Poisson brackets are also important in thermodynamics, see https://johncarlosbaez.wordpress.com/2012/01/23/classical-mechanics-versus-thermodynamics-part-2/ and M. J. Peterson, Analogy between thermodynamics and mechanics, American Journal of Physics 47 (1979), 488–490. |

<tabbox FAQ> | <tabbox FAQ> |

advanced_notions/poisson_bracket.1523187517.txt.gz · Last modified: 2018/04/08 11:38 (external edit)

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